Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is $$\alpha \times 5^6$$, then $$\alpha$$ is equal to

Passwords are 6 to 8 characters long. Each character is from {A, B, C, D, E, 1, 2, 3, 4, 5} (10 characters total), and we require at least one number. Therefore, the total number of passwords is $$10^6 + 10^7 + 10^8.$$

To exclude those with no numbers, we consider passwords using only the five letters {A, B, C, D, E}, which gives $$5^6 + 5^7 + 5^8. $$

Subtracting these all-letter passwords from the total yields $$= (10^6 + 10^7 + 10^8) - (5^6 + 5^7 + 5^8). $$

Rewriting the difference term by term, we have $$= (10^6 - 5^6) + (10^7 - 5^7) + (10^8 - 5^8). $$

Since $$10^n = (2 \cdot 5)^n = 2^n \cdot 5^n$$, we can factor each term as follows: $$= 5^6(2^6 - 1) + 5^7(2^7 - 1) + 5^8(2^8 - 1). $$

Grouping the factors, this becomes $$= 5^6[63 + 5 \times 127 + 25 \times 255], $$ so that $$= 5^6[63 + 635 + 6375]. $$

Therefore, we obtain $$= 5^6 \times 7073, $$ and hence $$\alpha = 7073. $$

The answer is 7073.